Good action of a nilpotent group with regular orbits

TL;DR

This paper investigates the action of finite nilpotent groups of odd order on other groups, establishing bounds on the Fitting height based on prime divisors, under specific conditions.

Contribution

It provides new bounds on the Fitting height of a group acted upon by a nilpotent group, extending understanding of group actions with regular orbits.

Findings

Fitting height is bounded by prime divisors of the acting group and centralizer.

Results apply to groups of odd order with 'good' actions.

Provides conditions under which these bounds hold.

Abstract

Suppose that $A$ is a finite nilpotent group of odd order acting good in the sense of \cite{EGJ} on the group $G$ of odd order. Under some additional assumptions we prove that the Fitting height of $G$ is bounded above by the sum of the numbers of primes dividing $|A|$ and $|C_G(A)|$ counted with multiplicities.